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G = C2.(C4×Q16)  order 128 = 27

6th central stem extension by C2 of C4×Q16

p-group, metabelian, nilpotent (class 3), monomial

Aliases: C4.69(C4×D4), C2.9(C4×Q16), C4⋊C4.214D4, Q8⋊C45C4, (C2×C4).49Q16, C2.3(C42Q16), (C22×C4).698D4, C23.783(C2×D4), C22.165(C4×D4), C22.35(C2×Q16), C2.5(D4.2D4), C4.27(C4.4D4), C22.68(C4○D8), (C22×C8).51C22, C4.32(C422C2), C4.39(C42⋊C2), C22.86(C8⋊C22), C22.4Q16.12C2, (C2×C42).299C22, C2.15(SD16⋊C4), (C22×Q8).28C22, C22.127(C4⋊D4), (C22×C4).1382C23, C2.7(C23.20D4), C2.5(C23.48D4), C22.75(C8.C22), C23.67C23.7C2, C22.7C42.22C2, C2.13(C24.C22), C22.95(C22.D4), (C4×C4⋊C4).19C2, (C2×C8).42(C2×C4), (C2×C2.D8).7C2, C4⋊C4.152(C2×C4), (C2×Q8).80(C2×C4), (C2×C4).1012(C2×D4), (C2×Q8⋊C4).5C2, (C2×C4).578(C4○D4), (C2×C4⋊C4).776C22, (C2×C4).400(C22×C4), SmallGroup(128,660)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C2.(C4×Q16)
C1C2C22C2×C4C22×C4C2×C4⋊C4C4×C4⋊C4 — C2.(C4×Q16)
C1C2C2×C4 — C2.(C4×Q16)
C1C23C2×C42 — C2.(C4×Q16)
C1C2C2C22×C4 — C2.(C4×Q16)

Generators and relations for C2.(C4×Q16)
 G = < a,b,c,d | a2=b4=c8=1, d2=c4, cbc-1=dbd-1=ab=ba, ac=ca, ad=da, dcd-1=ac-1 >

Subgroups: 244 in 127 conjugacy classes, 58 normal (44 characteristic)
C1, C2 [×7], C4 [×4], C4 [×10], C22 [×7], C8 [×3], C2×C4 [×6], C2×C4 [×2], C2×C4 [×20], Q8 [×6], C23, C42 [×4], C4⋊C4 [×4], C4⋊C4 [×4], C2×C8 [×2], C2×C8 [×5], C22×C4 [×3], C22×C4 [×4], C2×Q8 [×2], C2×Q8 [×5], C2.C42 [×3], Q8⋊C4 [×4], Q8⋊C4 [×2], C2.D8 [×2], C2×C42, C2×C42, C2×C4⋊C4 [×3], C22×C8 [×2], C22×Q8, C22.7C42, C22.4Q16, C4×C4⋊C4, C23.67C23, C2×Q8⋊C4 [×2], C2×C2.D8, C2.(C4×Q16)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, Q16 [×2], C22×C4, C2×D4 [×2], C4○D4 [×4], C42⋊C2, C4×D4 [×2], C4⋊D4, C22.D4, C4.4D4, C422C2, C2×Q16, C4○D8, C8⋊C22, C8.C22, C24.C22, C4×Q16, SD16⋊C4, C42Q16, D4.2D4, C23.48D4, C23.20D4, C2.(C4×Q16)

Smallest permutation representation of C2.(C4×Q16)
Regular action on 128 points
Generators in S128
(1 128)(2 121)(3 122)(4 123)(5 124)(6 125)(7 126)(8 127)(9 107)(10 108)(11 109)(12 110)(13 111)(14 112)(15 105)(16 106)(17 28)(18 29)(19 30)(20 31)(21 32)(22 25)(23 26)(24 27)(33 47)(34 48)(35 41)(36 42)(37 43)(38 44)(39 45)(40 46)(49 60)(50 61)(51 62)(52 63)(53 64)(54 57)(55 58)(56 59)(65 118)(66 119)(67 120)(68 113)(69 114)(70 115)(71 116)(72 117)(73 88)(74 81)(75 82)(76 83)(77 84)(78 85)(79 86)(80 87)(89 100)(90 101)(91 102)(92 103)(93 104)(94 97)(95 98)(96 99)
(1 30 114 78)(2 20 115 86)(3 32 116 80)(4 22 117 88)(5 26 118 74)(6 24 119 82)(7 28 120 76)(8 18 113 84)(9 98 52 39)(10 96 53 46)(11 100 54 33)(12 90 55 48)(13 102 56 35)(14 92 49 42)(15 104 50 37)(16 94 51 44)(17 67 83 126)(19 69 85 128)(21 71 87 122)(23 65 81 124)(25 72 73 123)(27 66 75 125)(29 68 77 127)(31 70 79 121)(34 110 101 58)(36 112 103 60)(38 106 97 62)(40 108 99 64)(41 111 91 59)(43 105 93 61)(45 107 95 63)(47 109 89 57)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)(65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88)(89 90 91 92 93 94 95 96)(97 98 99 100 101 102 103 104)(105 106 107 108 109 110 111 112)(113 114 115 116 117 118 119 120)(121 122 123 124 125 126 127 128)
(1 62 5 58)(2 50 6 54)(3 60 7 64)(4 56 8 52)(9 117 13 113)(10 71 14 67)(11 115 15 119)(12 69 16 65)(17 40 21 36)(18 45 22 41)(19 38 23 34)(20 43 24 47)(25 35 29 39)(26 48 30 44)(27 33 31 37)(28 46 32 42)(49 126 53 122)(51 124 55 128)(57 121 61 125)(59 127 63 123)(66 109 70 105)(68 107 72 111)(73 102 77 98)(74 90 78 94)(75 100 79 104)(76 96 80 92)(81 101 85 97)(82 89 86 93)(83 99 87 103)(84 95 88 91)(106 118 110 114)(108 116 112 120)

G:=sub<Sym(128)| (1,128)(2,121)(3,122)(4,123)(5,124)(6,125)(7,126)(8,127)(9,107)(10,108)(11,109)(12,110)(13,111)(14,112)(15,105)(16,106)(17,28)(18,29)(19,30)(20,31)(21,32)(22,25)(23,26)(24,27)(33,47)(34,48)(35,41)(36,42)(37,43)(38,44)(39,45)(40,46)(49,60)(50,61)(51,62)(52,63)(53,64)(54,57)(55,58)(56,59)(65,118)(66,119)(67,120)(68,113)(69,114)(70,115)(71,116)(72,117)(73,88)(74,81)(75,82)(76,83)(77,84)(78,85)(79,86)(80,87)(89,100)(90,101)(91,102)(92,103)(93,104)(94,97)(95,98)(96,99), (1,30,114,78)(2,20,115,86)(3,32,116,80)(4,22,117,88)(5,26,118,74)(6,24,119,82)(7,28,120,76)(8,18,113,84)(9,98,52,39)(10,96,53,46)(11,100,54,33)(12,90,55,48)(13,102,56,35)(14,92,49,42)(15,104,50,37)(16,94,51,44)(17,67,83,126)(19,69,85,128)(21,71,87,122)(23,65,81,124)(25,72,73,123)(27,66,75,125)(29,68,77,127)(31,70,79,121)(34,110,101,58)(36,112,103,60)(38,106,97,62)(40,108,99,64)(41,111,91,59)(43,105,93,61)(45,107,95,63)(47,109,89,57), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104)(105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128), (1,62,5,58)(2,50,6,54)(3,60,7,64)(4,56,8,52)(9,117,13,113)(10,71,14,67)(11,115,15,119)(12,69,16,65)(17,40,21,36)(18,45,22,41)(19,38,23,34)(20,43,24,47)(25,35,29,39)(26,48,30,44)(27,33,31,37)(28,46,32,42)(49,126,53,122)(51,124,55,128)(57,121,61,125)(59,127,63,123)(66,109,70,105)(68,107,72,111)(73,102,77,98)(74,90,78,94)(75,100,79,104)(76,96,80,92)(81,101,85,97)(82,89,86,93)(83,99,87,103)(84,95,88,91)(106,118,110,114)(108,116,112,120)>;

G:=Group( (1,128)(2,121)(3,122)(4,123)(5,124)(6,125)(7,126)(8,127)(9,107)(10,108)(11,109)(12,110)(13,111)(14,112)(15,105)(16,106)(17,28)(18,29)(19,30)(20,31)(21,32)(22,25)(23,26)(24,27)(33,47)(34,48)(35,41)(36,42)(37,43)(38,44)(39,45)(40,46)(49,60)(50,61)(51,62)(52,63)(53,64)(54,57)(55,58)(56,59)(65,118)(66,119)(67,120)(68,113)(69,114)(70,115)(71,116)(72,117)(73,88)(74,81)(75,82)(76,83)(77,84)(78,85)(79,86)(80,87)(89,100)(90,101)(91,102)(92,103)(93,104)(94,97)(95,98)(96,99), (1,30,114,78)(2,20,115,86)(3,32,116,80)(4,22,117,88)(5,26,118,74)(6,24,119,82)(7,28,120,76)(8,18,113,84)(9,98,52,39)(10,96,53,46)(11,100,54,33)(12,90,55,48)(13,102,56,35)(14,92,49,42)(15,104,50,37)(16,94,51,44)(17,67,83,126)(19,69,85,128)(21,71,87,122)(23,65,81,124)(25,72,73,123)(27,66,75,125)(29,68,77,127)(31,70,79,121)(34,110,101,58)(36,112,103,60)(38,106,97,62)(40,108,99,64)(41,111,91,59)(43,105,93,61)(45,107,95,63)(47,109,89,57), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104)(105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128), (1,62,5,58)(2,50,6,54)(3,60,7,64)(4,56,8,52)(9,117,13,113)(10,71,14,67)(11,115,15,119)(12,69,16,65)(17,40,21,36)(18,45,22,41)(19,38,23,34)(20,43,24,47)(25,35,29,39)(26,48,30,44)(27,33,31,37)(28,46,32,42)(49,126,53,122)(51,124,55,128)(57,121,61,125)(59,127,63,123)(66,109,70,105)(68,107,72,111)(73,102,77,98)(74,90,78,94)(75,100,79,104)(76,96,80,92)(81,101,85,97)(82,89,86,93)(83,99,87,103)(84,95,88,91)(106,118,110,114)(108,116,112,120) );

G=PermutationGroup([(1,128),(2,121),(3,122),(4,123),(5,124),(6,125),(7,126),(8,127),(9,107),(10,108),(11,109),(12,110),(13,111),(14,112),(15,105),(16,106),(17,28),(18,29),(19,30),(20,31),(21,32),(22,25),(23,26),(24,27),(33,47),(34,48),(35,41),(36,42),(37,43),(38,44),(39,45),(40,46),(49,60),(50,61),(51,62),(52,63),(53,64),(54,57),(55,58),(56,59),(65,118),(66,119),(67,120),(68,113),(69,114),(70,115),(71,116),(72,117),(73,88),(74,81),(75,82),(76,83),(77,84),(78,85),(79,86),(80,87),(89,100),(90,101),(91,102),(92,103),(93,104),(94,97),(95,98),(96,99)], [(1,30,114,78),(2,20,115,86),(3,32,116,80),(4,22,117,88),(5,26,118,74),(6,24,119,82),(7,28,120,76),(8,18,113,84),(9,98,52,39),(10,96,53,46),(11,100,54,33),(12,90,55,48),(13,102,56,35),(14,92,49,42),(15,104,50,37),(16,94,51,44),(17,67,83,126),(19,69,85,128),(21,71,87,122),(23,65,81,124),(25,72,73,123),(27,66,75,125),(29,68,77,127),(31,70,79,121),(34,110,101,58),(36,112,103,60),(38,106,97,62),(40,108,99,64),(41,111,91,59),(43,105,93,61),(45,107,95,63),(47,109,89,57)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64),(65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88),(89,90,91,92,93,94,95,96),(97,98,99,100,101,102,103,104),(105,106,107,108,109,110,111,112),(113,114,115,116,117,118,119,120),(121,122,123,124,125,126,127,128)], [(1,62,5,58),(2,50,6,54),(3,60,7,64),(4,56,8,52),(9,117,13,113),(10,71,14,67),(11,115,15,119),(12,69,16,65),(17,40,21,36),(18,45,22,41),(19,38,23,34),(20,43,24,47),(25,35,29,39),(26,48,30,44),(27,33,31,37),(28,46,32,42),(49,126,53,122),(51,124,55,128),(57,121,61,125),(59,127,63,123),(66,109,70,105),(68,107,72,111),(73,102,77,98),(74,90,78,94),(75,100,79,104),(76,96,80,92),(81,101,85,97),(82,89,86,93),(83,99,87,103),(84,95,88,91),(106,118,110,114),(108,116,112,120)])

38 conjugacy classes

class 1 2A···2G4A···4H4I···4R4S4T4U4V8A···8H
order12···24···44···444448···8
size11···12···24···488884···4

38 irreducible representations

dim111111112222244
type+++++++++-+-
imageC1C2C2C2C2C2C2C4D4D4Q16C4○D4C4○D8C8⋊C22C8.C22
kernelC2.(C4×Q16)C22.7C42C22.4Q16C4×C4⋊C4C23.67C23C2×Q8⋊C4C2×C2.D8Q8⋊C4C4⋊C4C22×C4C2×C4C2×C4C22C22C22
# reps111112182248411

Matrix representation of C2.(C4×Q16) in GL6(𝔽17)

100000
010000
0016000
0001600
000010
000001
,
1300000
0130000
0013800
0013400
0000160
0000016
,
0110000
300000
0011500
0001600
0000611
000030
,
1170000
1260000
0016000
0016100
000040
0000413

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,13,13,0,0,0,0,8,4,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[0,3,0,0,0,0,11,0,0,0,0,0,0,0,1,0,0,0,0,0,15,16,0,0,0,0,0,0,6,3,0,0,0,0,11,0],[11,12,0,0,0,0,7,6,0,0,0,0,0,0,16,16,0,0,0,0,0,1,0,0,0,0,0,0,4,4,0,0,0,0,0,13] >;

C2.(C4×Q16) in GAP, Magma, Sage, TeX

C_2.(C_4\times Q_{16})
% in TeX

G:=Group("C2.(C4xQ16)");
// GroupNames label

G:=SmallGroup(128,660);
// by ID

G=gap.SmallGroup(128,660);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,736,422,394,2804,718,172]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^4=c^8=1,d^2=c^4,c*b*c^-1=d*b*d^-1=a*b=b*a,a*c=c*a,a*d=d*a,d*c*d^-1=a*c^-1>;
// generators/relations

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